Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

A one to one correspondence between shifts of group-like projections on a locally compact quantum group ${\mathbb{G}}$ which are preserved by the scaling group and contractive idempotent functionals on the dual $\hat{\mathbb{G}}$ is established. This is a generalization of the Illie-Spronk's correspondence between contractive idempotents in the Fourier-Stieltjes algebra of a locally compact group $G$ and cosets of open subgroups of $G$. We also establish a one to one correspondence between non-degenerate, integrable, ${\mathbb{G}}$-invariant ternary rings of operators $X\subset L^\infty({\mathbb{G}})$, preserved by the scaling group and contractive idempotent functionals on ${\mathbb{G}}$. Using our results we characterize coideals in $L^\infty(\hat{\mathbb{G}})$ admitting an atom preserved by the scaling group in terms of idempotent states on ${\mathbb{G}}$. We also establish a one to one correspondence between integrable coideals in $L^\infty({\mathbb{G}})$ and group-like projections in $L^\infty(\hat{\mathbb{G}})$ satisfying an extra mild condition. Exploiting this correspondence we give examples of group like projections which are not preserved by the scaling group.